Euclid, Elements
Thomas L. Heath, Sir Thomas Little Heath, Ed.

For the same reason F is also square.

Similarly we can prove that all those which leave out one are square.

I say next that C, the fourth from the unit, is cube, as are also all those which leave out two.

For since, as the unit is to A, so is B to C, therefore the unit measures the number A the same number of times that B measures C.

But the unit measures the number A according to the units in A; therefore B also measures C according to the units in A.

Therefore A by multiplying B has made C.

Since then A by multiplying itself has made B, and by multiplying B has made C, therefore C is cube.

And, since C, D, E, F are in continued proportion, and C is cube, therefore F is also cube. [VIII. 23]

But it was also proved square; therefore the seventh from the unit is both cube and square.

Similarly we can prove that all the numbers which leave out five are also both cube and square. Q. E. D.

PROPOSITION 9.

Let there be as many numbers as we please, A, B, C, D, E, F, beginning from an unit and in continued proportion, and let A, the number after the unit, be square; I say that all the rest will also be square.

Now it has been proved that B, the third from the unit, is square, as are also all those which leave out one; [IX. 8] I say that all the rest are also square.

For, since A, B, C are in continued proportion, and A is square, therefore C is also square. [VIII. 22]

Again, since B, C, D are in continued proportion, and B is square, D is also square. [VIII. 22]

Similarly we can prove that all the rest are also square.

Next, let A be cube; I say that all the rest are also cube.

Now it has been proved that C, the fourth from the unit, is cube, as also are all those which leave out two; [IX. 8] I say that all the rest are also cube.

For, since, as the unit is to A, so is A to B, therefore the unit measures A the same number of times as A measures B.

But the unit measures A according to the units in it; therefore A also measures B according to the units in itself; therefore A by multiplying itself has made B.